Most physicists now seem to think that the answer to the big questions about the universe is found in eleven dimensions, in something called M-theory. But to me it is very heartening that some mavericks are still prepared to look at other beautiful objects, in the hope that God wouldn’t have been able to resist using them in his construction.
Among these are:
- the octonions, an 8-dimensional alternative algebra (this means that it is not associative, but satisfies a restricted version of the associative law);
- the 27-dimensional exceptional Jordan algebra, which is built from 3×3 Hermitian matrices over the octonions;
- the 248-dimensional exceptional Lie algebra E8, the largest of the five exceptional Lie algebras;
- the 196884-dimensional Griess algebra, used to construct the Monster (and used by Borcherds to construct an infinite-dimensional Lie algebra in his proof of the Moonshine conjectures).
So it was very nice that, in the London algebra colloquium last week, there was a talk by Rob Wilson which wove together the first three of these, describing joint work with two physicists (whose names I am afraid I didn’t catch, and can’t find on the web).
He began with Freudenthal’s magic square, a 4×4 square whose entries include the exceptional Lie algebras F4, E6, E7 and E8, as well as some of the more interesting classical ones. Freudental’s square has rows labelled by four different kinds of geometry (elliptic, hyperbolic, symplectic, and metasymplectic), and columns by four division algebras (real numbers, complex numbers, quaternions, and octonions); part of its magic is that it turns out to be symmetric!
The explanation turns out to be that the algebras can be built in a uniform way from 3×3 matrices but over the tensor product of two of the division algebras; so the rows and columns are both labelled by division algebras, and the symmetry of the table follows from that of the tensor product.
However, great care is required. We have to use actions of the matrices on suitable spaces rather than the matrices themselves. Now actions lead naturally to non-commutativity (putting on your shoes first and then your socks leads to quite a different result from the usual procedure), but not to non-associativity, since both (ab)c and a(bc) mean “first a, then b, then c“. So if the rules coming from the action conflict with the usual rules for matrix multiplication, they take precedence; this means it is fatally easy to get signs wrong!
Since the main construction uses O⊗O, he describes it as “cooking up E8 with an Oxo cube”.
More comes out of this. By using different real forms of the division algebras, one can get different things out of the construction; Rob hopes to tweak the E8 construction to get Thompson’s simple group.